Fixed and rear brake only (i.e. no front brake) isn't legal, if that's a concern for you. Fixed and front brake only is legal. Singlespeed requires both brakes to be legal.
Personally I wouldn't want a single brake on fixed, I've had a brake cable snap before and wouldn't want to be left having to rely on leg braking alone but then I use my fixed for long hilly rides. You may decide the risk of a single brake (or even none at all) is acceptable for blatting around flatish roads.
Fixed and rear brake only (i.e. no front brake) isn't legal, if that's a concern for you. Fixed and front brake only is legal. Singlespeed requires both brakes to be legal.
Personally I wouldn't want a single brake on fixed, I've had a brake cable snap before and wouldn't want to be left having to rely on leg braking alone but then I use my fixed for long hilly rides. You may decide the risk of a single brake (or even none at all) is acceptable for blatting around flatish roads.